Simplify The Rational Expression.$\frac{y^2+6y+8}{2y^2+y-6} \cdot \frac{2y^2+7y-15}{y^2-16} =$

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying a rational expression, using the given expression as an example. We will break down the expression into smaller parts, factorize the numerator and denominator, and then simplify the resulting expression.

The Given Expression

The given rational expression is:

y2+6y+82y2+y−6⋅2y2+7y−15y2−16\frac{y^2+6y+8}{2y^2+y-6} \cdot \frac{2y^2+7y-15}{y^2-16}

Step 1: Factorize the Numerator and Denominator

To simplify the rational expression, we need to factorize the numerator and denominator. Let's start with the numerator:

y2+6y+82y2+y−6⋅2y2+7y−15y2−16\frac{y^2+6y+8}{2y^2+y-6} \cdot \frac{2y^2+7y-15}{y^2-16}

The numerator can be factorized as:

y2+6y+8=(y+4)(y+2)y^2+6y+8 = (y+4)(y+2)

The denominator can be factorized as:

2y2+y−6=(2y−3)(y+2)2y^2+y-6 = (2y-3)(y+2)

y2−16=(y−4)(y+4)y^2-16 = (y-4)(y+4)

Step 2: Cancel Out Common Factors

Now that we have factorized the numerator and denominator, we can cancel out common factors. The expression becomes:

(y+4)(y+2)(2y−3)(y+2)⋅(2y+5)(y−3)(y−4)(y+4)\frac{(y+4)(y+2)}{(2y-3)(y+2)} \cdot \frac{(2y+5)(y-3)}{(y-4)(y+4)}

We can cancel out the common factor (y+2)(y+2) from the numerator and denominator:

(y+4)(2y−3)⋅(2y+5)(y−3)(y−4)(y+4)\frac{(y+4)}{(2y-3)} \cdot \frac{(2y+5)(y-3)}{(y-4)(y+4)}

Step 3: Simplify the Expression

Now that we have canceled out the common factors, we can simplify the expression further. We can cancel out the common factor (y+4)(y+4) from the numerator and denominator:

1(2y−3)⋅(2y+5)(y−3)(y−4)\frac{1}{(2y-3)} \cdot \frac{(2y+5)(y-3)}{(y-4)}

Step 4: Multiply the Numerator and Denominator

Finally, we can multiply the numerator and denominator to get the simplified expression:

(2y+5)(y−3)(2y−3)(y−4)\frac{(2y+5)(y-3)}{(2y-3)(y-4)}

Conclusion

Simplifying a rational expression involves factorizing the numerator and denominator, canceling out common factors, and multiplying the remaining factors. By following these steps, we can simplify the given rational expression and arrive at the final answer.

Final Answer

The simplified rational expression is:

(2y+5)(y−3)(2y−3)(y−4)\frac{(2y+5)(y-3)}{(2y-3)(y-4)}

Tips and Tricks

  • Factorize the numerator and denominator before canceling out common factors.
  • Cancel out common factors to simplify the expression.
  • Multiply the remaining factors to get the final answer.

Real-World Applications

Simplifying rational expressions has many real-world applications, including:

  • Calculating probabilities and statistics
  • Modeling population growth and decay
  • Analyzing financial data and making investment decisions

Common Mistakes

  • Failing to factorize the numerator and denominator
  • Canceling out common factors incorrectly
  • Not multiplying the remaining factors

Practice Problems

  • Simplify the rational expression: x2+5x+6x2+3x−2⋅x2+2x−3x2−9\frac{x^2+5x+6}{x^2+3x-2} \cdot \frac{x^2+2x-3}{x^2-9}
  • Simplify the rational expression: y2+2y−3y2+5y+6⋅y2+3y−2y2−4\frac{y^2+2y-3}{y^2+5y+6} \cdot \frac{y^2+3y-2}{y^2-4}

References

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Introduction

In our previous article, we explored the process of simplifying a rational expression. We broke down the expression into smaller parts, factorized the numerator and denominator, and then simplified the resulting expression. In this article, we will answer some frequently asked questions about simplifying rational expressions.

Q: What is a rational expression?

A rational expression is a mathematical expression that consists of a fraction, where the numerator and denominator are polynomials.

A: How do I simplify a rational expression?

To simplify a rational expression, you need to follow these steps:

  1. Factorize the numerator and denominator.
  2. Cancel out common factors.
  3. Multiply the remaining factors.

Q: What is the difference between simplifying and canceling?

Simplifying a rational expression involves breaking down the expression into smaller parts, factorizing the numerator and denominator, and then multiplying the remaining factors. Canceling out common factors is a step in the simplification process.

Q: How do I factorize a polynomial?

To factorize a polynomial, you need to find the greatest common factor (GCF) of the terms and then factor out the GCF. You can also use the factoring method, which involves finding two binomials whose product is the original polynomial.

Q: What is the greatest common factor (GCF)?

The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers without leaving a remainder.

Q: How do I cancel out common factors?

To cancel out common factors, you need to identify the common factors in the numerator and denominator and then cancel them out.

Q: What are some common mistakes to avoid when simplifying rational expressions?

Some common mistakes to avoid when simplifying rational expressions include:

  • Failing to factorize the numerator and denominator
  • Canceling out common factors incorrectly
  • Not multiplying the remaining factors

Q: How do I know if a rational expression is already simplified?

A rational expression is already simplified if there are no common factors that can be canceled out.

Q: Can I simplify a rational expression with a variable in the denominator?

Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful not to cancel out the variable.

Q: How do I simplify a rational expression with a negative exponent?

To simplify a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent and then simplify.

Q: Can I simplify a rational expression with a fraction in the denominator?

Yes, you can simplify a rational expression with a fraction in the denominator. However, you need to be careful not to cancel out the fraction.

Q: How do I simplify a rational expression with a complex number in the denominator?

To simplify a rational expression with a complex number in the denominator, you need to use the conjugate of the complex number to rationalize the denominator.

Q: What are some real-world applications of simplifying rational expressions?

Simplifying rational expressions has many real-world applications, including:

  • Calculating probabilities and statistics
  • Modeling population growth and decay
  • Analyzing financial data and making investment decisions

Conclusion

Simplifying rational expressions is an essential skill in algebra and has many real-world applications. By following the steps outlined in this article, you can simplify rational expressions and arrive at the final answer.

Final Tips

  • Factorize the numerator and denominator before canceling out common factors.
  • Cancel out common factors to simplify the expression.
  • Multiply the remaining factors to get the final answer.

Practice Problems

  • Simplify the rational expression: x2+5x+6x2+3x−2⋅x2+2x−3x2−9\frac{x^2+5x+6}{x^2+3x-2} \cdot \frac{x^2+2x-3}{x^2-9}
  • Simplify the rational expression: y2+2y−3y2+5y+6⋅y2+3y−2y2−4\frac{y^2+2y-3}{y^2+5y+6} \cdot \frac{y^2+3y-2}{y^2-4}

References